Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1150,4,Mod(1,1150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1150.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1150 = 2 \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1150.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(67.8521965066\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 68x^{2} - 111x + 342 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 230) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 68x^{2} - 111x + 342 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 3\nu^{2} - 50\nu + 18 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 2\nu - 34 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2\beta _1 + 34 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 3\beta_{2} + 56\beta _1 + 84 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
2.00000 | −7.73081 | 4.00000 | 0 | −15.4616 | −23.5622 | 8.00000 | 32.7654 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −0.589969 | 4.00000 | 0 | −1.17994 | 18.5077 | 8.00000 | −26.6519 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 4.74869 | 4.00000 | 0 | 9.49738 | −29.3684 | 8.00000 | −4.44993 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 7.57209 | 4.00000 | 0 | 15.1442 | 35.4229 | 8.00000 | 30.3365 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(23\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1150.4.a.p | 4 | |
| 5.b | even | 2 | 1 | 230.4.a.h | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 1150.4.b.n | 8 | ||
| 15.d | odd | 2 | 1 | 2070.4.a.bj | 4 | ||
| 20.d | odd | 2 | 1 | 1840.4.a.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.4.a.h | ✓ | 4 | 5.b | even | 2 | 1 | |
| 1150.4.a.p | 4 | 1.a | even | 1 | 1 | trivial | |
| 1150.4.b.n | 8 | 5.c | odd | 4 | 2 | ||
| 1840.4.a.m | 4 | 20.d | odd | 2 | 1 | ||
| 2070.4.a.bj | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1150))\):
|
\( T_{3}^{4} - 4T_{3}^{3} - 62T_{3}^{2} + 243T_{3} + 164 \)
|
|
\( T_{7}^{4} - T_{7}^{3} - 1507T_{7}^{2} - 2618T_{7} + 453664 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 164 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - T^{3} + \cdots + 453664 \)
|
| $11$ |
\( T^{4} + 39 T^{3} + \cdots - 977536 \)
|
| $13$ |
\( T^{4} - 20 T^{3} + \cdots + 3059002 \)
|
| $17$ |
\( T^{4} - 23 T^{3} + \cdots + 1048184 \)
|
| $19$ |
\( T^{4} - 53 T^{3} + \cdots - 78336 \)
|
| $23$ |
\( (T - 23)^{4} \)
|
| $29$ |
\( T^{4} - 161 T^{3} + \cdots + 1597064 \)
|
| $31$ |
\( T^{4} - 388 T^{3} + \cdots - 397027152 \)
|
| $37$ |
\( T^{4} + 466 T^{3} + \cdots - 7921024 \)
|
| $41$ |
\( T^{4} + \cdots + 1506099394 \)
|
| $43$ |
\( T^{4} + \cdots - 5392023552 \)
|
| $47$ |
\( T^{4} + \cdots + 1306137888 \)
|
| $53$ |
\( T^{4} + \cdots - 3236491568 \)
|
| $59$ |
\( T^{4} + \cdots + 11673423616 \)
|
| $61$ |
\( T^{4} + \cdots - 42772329400 \)
|
| $67$ |
\( T^{4} + \cdots - 10308616192 \)
|
| $71$ |
\( T^{4} + \cdots + 274201266224 \)
|
| $73$ |
\( T^{4} + \cdots - 4754107544 \)
|
| $79$ |
\( T^{4} + \cdots + 145785325568 \)
|
| $83$ |
\( T^{4} + \cdots - 178673654272 \)
|
| $89$ |
\( T^{4} + \cdots - 969417760000 \)
|
| $97$ |
\( T^{4} + \cdots - 658266694276 \)
|
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